(s 2 - k 2 ) Y^s) - e iBl [y'a-) - is y(t-)] 



- [y'(0+) - is y(0+)] 



- - (s 2 - k 2 ) Y (s) 

 o + 



" (** ~ k o } Y - (s) 



+ 1(8 - k Q ) 



on use of (7.15), (7.2), and (7.7), so that 



i 



Y + (s) 4 Y_(s) + K(s) Y t (s) - r 



where the kernel l.s, as in §2 t 

 s 2 - k 2 

 2 - k 2 



K-(s) 



(7.18) 



(7.19) 



This Is the required generalized W-H equation, a single equation for 



three unknown functions Y (s), Y (s) and Yj(s), given the kernel K(s) 



and the forcing field (s + k ) _1 . 



o 



Equations of this kind have been considered by Noble [5 p. 196]. 

 Methods exist for solving such equations approximately in "high 

 frequency" limits in which the finite part of the boundary is many 

 wavelengths long, in some appropriate sense. Generally these methods 

 rely on weak interaction between the ends x « and x » I, though as 

 remarked before, the ends are rather delicately coupled in cases such 

 as the resonance of a finite open-ended waveguide, and nethcds have 

 also been developed to deal with such cases. Here, because only pole 

 singularities are involved, it is pocsible to solve (7.18) exactly. 



39 



